Resistance Coefficients for Non-Newtonian Flows in Pipe Fittings

The focus of this chapter is to provide a review of the loss coefficient data for laminar flow of non-Newtonian fluids in pipe fittings. Since the total pressure change in a piping system generally consists of three components: (i) the frictional pressure loss in the pipe, (ii) the frictional pressure loss arising from flow through fittings and (iii) the pressure loss or gain resulting from elevation changes, this review will also deal with laminar and turbulent pipe flow of non-Newtonian fluids and the application of viscometry for flow in pipes and fittings. The rheological models relevant to industrial fluids such as mine tailings and sewage sludges are introduced, with particular emphasis on yield stress, or viscoplastic, fluids.


Introduction
The focus of this chapter is to provide a review of the loss coefficient data for laminar flow of non-Newtonian fluids in pipe fittings. Since the total pressure change in a piping system generally consists of three components: (i) the frictional pressure loss in the pipe, (ii) the frictional pressure loss arising from flow through fittings and (iii) the pressure loss or gain resulting from elevation changes, this review will also deal with laminar and turbulent pipe flow of non-Newtonian fluids and the application of viscometry for flow in pipes and fittings. The rheological models relevant to industrial fluids such as mine tailings and sewage sludges are introduced, with particular emphasis on yield stress, or viscoplastic, fluids. Hooper (1981) presented a two-K method for determining the loss coefficient for laminar and turbulent flow through various fittings and valves. This method consists of two factors, one for laminar flow, K 1 and the other for turbulent flow, K turb . Unlike that for K turb , there is little data available for K 1 . Experimental data over the full range of laminar and turbulent flow are presented for flow of Newtonian and non-Newtonian fluids in various fittings. The experimental procedures for the accurate determination of loss coefficients are described.
Current practice for laminar flow through various fittings is to present the loss coefficient as a function of an appropriate Reynolds number. Different Reynolds numbers developed for non-Newtonian fluids have been evaluated to determine their ability to establish the necessary requirement of dynamic similarity for flow of viscoplastic fluids in various fittings.
The laminar to turbulent transition in pipe fittings are also discussed. The experimental work done to date on contractions, expansions, valves and orifices is reviewed in addition to similar work published in literature.
The magnitude of errors that can be obtained using the incorrect loss coefficient is demonstrated by means of a worked example. This chapter will provide the pipeline design www.intechopen.com engineer dealing with non-Newtonian fluids with the necessary information critical for energy efficient design.

Rheological models
The main types of flow behaviour exhibited by fluids under steady-state shear include: • Newtonian • non-Newtonian • shear-thinning and dilatant (shear-thickening) • viscoplastic Over a limited shear rate range (of one or two decades), a fluid can exhibit a single class of behaviour characterised by the flow curves of Sections 2.1, 2.2.1 or 2.2.2. Over a wider shear rate range (within 10 -6 to 10 6 s -1 ), most fluids exhibit more than one class of flow behaviour.
It is difficult to predict the type of flow behaviour that a fluid will exhibit under given flow conditions (for example, at a given temperature, pressure and concentration). Nevertheless, there are textbooks such as those of Laba (1993) and Steffe (1996) that give examples that can serve as illustrations only. The user should be aware that a rheological test is the only sure method of ascertaining the rheological behaviour of a fluid (Alderman, 1997).

Newtonian behaviour
These are fluids for which an infinitesimal shear stress will initiate flow and for which the shear stress is directly proportional to the shear rate. The flow curve, at a given temperature and pressure, is therefore linear and passes through the origin as is shown in Figure 1(a). The slope of the flow curve, which is constant, is the viscosity. Re-plotting the flow curve in the form of a viscosity curve as shown in Figure 1(b) clearly depicts a constant viscosity with respect to shear rate. (1)

Fig. 1. Flow and viscosity curves for fluids exhibiting Newtonian behaviour
Any deviation from Newtonian behaviour is said to be non-Newtonian.
Shear-thinning behaviour is observed when an infinitesimal shear stress will initiate flow, that is, the flow curve passes through the origin, Figure 2(a) and the viscosity decreases with increasing shear rate as shown in Figure 2(b). This behaviour is sometimes incorrectly termed thixotropy because the equilibrium flow curve of a thixotropic material is often shear-thinning. However, unlike shear-thinning behaviour, thixotropy is a time-dependent property. Dilatant (shear-thickening) behaviour is observed when an infinitesimal shear stress will initiate flow, that is, the flow curve passes through the origin, Figure 2(a) and the viscosity increases with increasing shear rate as shown in Figure 2(b).

Fig. 2. Idealised shear-thinning and dilatant (shear-thickening) behaviour
For polymeric systems of low concentrations, low molecular weights or at temperatures well above the glass transition temperature, the variation of viscosity with shear rate as a function of concentration, molecular weight or temperature is shown typically in Figure 3. Fig. 3. Non-Newtonian viscosity of polymeric systems of low concentrations, low molecular weight or at temperatures well above the glass transition temperature (Alderman, 1997) It can be seen that as the concentration or molecular weight is increased or the temperature is decreased, both the zero and infinite shear viscosities, η 0 and η ∞ increases with the change in the zero shear viscosity being much greater than that for infinite shear viscosity. The www.intechopen.com difference between η 0 and η ∞ c a n b e v e r y l a r g e , a s m u c h a s t w o o r t h r e e o r d e r s o f magnitude even, if the molecular weight or concentration is sufficiently high. The characteristic shear rate, where the curve starts to deviate from the η 0 line, decreases with increasing concentration or molecular weight or with decreasing temperature. The characteristic shear rate, where the curve starts to deviate from the η 0 line, decreases with increasing concentration or molecular weight or with decreasing temperature.
For particulate systems containing a dispersed phase having negligible inter-particle attraction, the variation of viscosity with shear rate as a function of solids content, shown typically in Figure 4, is complex. At low solids concentrations, the curve depicts shearthinning behaviour with well-defined zero and infinite shear viscosities. The lower boundary of the shear-thinning region can be identified as the curve depicted in Figure 4. At higher solids concentrations, dilatant behaviour is observed at high shear rates. This becomes more pronounced with increasing solids concentration. The characteristic shear rate decreases with increasing concentration until at very high concentrations it falls outside the lowest measurable shear rate. The magnitude of the slope at the characteristic shear rate also increases until at some concentration, it attains a value of -1. If replotted as a flow curve, this would show a yield stress. Here, the flow behaviour is viscoplastic. Fig. 4. Non-Newtonian viscosity in suspensions of negligible inter-particle attraction (Alderman, 1997)

Viscoplastic behavior
Two further departures from Newtonian behaviour, namely Bingham plastic and viscoplastic b e h a v i o u r , a r e d e p i c t e d i n F i g u r e 5 . F l u i d s e x h i b i t i n g v i s c o p l a s t i c i t y w i l l sustain a certain shear stress, the yield stress, τ y , without developing continuous flow. For stresses below τ y , the shear rate remains zero whereas for stresses above τ y , the fluid flows with a shear rate dependent on the excess stress (τ -τ y ). Bingham plastic behaviour is observed when there is a linear relationship between the shear stress in excess of the yield stress, τ y and the resulting shear rate. The flow curve for a Bingham plastic fluid is linear but does not pass through the origin as shown in Figure 5(a). Viscoplastic behaviour is observed when the rate of increase in shear stress with shear rate in excess of the yield stress, τ y decreases with increasing shear rate. The flow curve, shown in Figure 5(a), has the same characteristic shape as that for a fluid exhibiting shear-thinning behaviour but does not pass through the origin. The flow curve for both fluids cuts the shear stress-axis above the origin at τ y on the linear plot, Figure 5(a). The viscosity for both fluids decreases with shear rate similar to that for a shear-thinning fluid, Figure 5(b). However, unlike the shear-thinning viscosity which is finite at zero shear rate, the Bingham plastic or viscoplastic viscosity tends to infinity as shear rate is reduced to zero.
Both polymeric and particulate systems can possess a yield stress. This occurs particularly for cross-linked polymers, polymeric gels and filled systems, and for high solids content suspensions and high dispersed phase emulsions and foams. In the case of suspensions, the yield stress arises because of particle-particle frictional interaction. In these materials, it is found that the Bingham plastic behaviour is obtained at high shear rates, as an asymptotic behaviour, Figure 5(a). The intercept of the extrapolated Bingham plastic asymptote is commonly called the Bingham yield stress, τ yB , to distinguish it from the yield stress, τ y .

Viscosity and flow curve models
Fluids exhibiting Newtonian behaviour have constant viscosities as defined by Eqn.
(1), and single point measurement at a convenient shear rate is sufficient to describe such a fluid. However, non-Newtonian fluids have viscosities that are shear rate-dependent, and a single point measurement is inadequate to describe the flow behaviour. Here, the relevant shear rate range in the engineering application must be assessed and used in determining the measurement conditions for the viscometer/rheometer. Shear ratedependent viscosity is often referred to as apparent viscosity, η a , but there is no need to make this distinction if it is accepted that viscosity, as defined in Eqn.
(1), can be variable. However, it is essential that values of η are quoted with their corresponding values of shear rate (or shear stress).
The flow curve or viscosity curve data obtained from viscometric measurements under steady-state shear can be described mathematically in terms of rheological models (constitutive equations) and are amenable to curve fitting. The general form of a rheological model is fitted over a relevant shear rate range to the flow curve or the viscosity curve by least-squares regression analysis so that a specific rheological model can be obtained. This specific equation is then used for material characterization, engineering design applications or product formulation.
The rheological model for Newtonian fluids contains just one constant, η N . Many models have been proposed to describe the non-Newtonian flow behaviour of fluids, although the majority of these are of little value for engineering design applications and serve more as theoretical analyses.
In general, the simpler models (for example, Newtonian, Bingham plastic and power law) are used for fitting narrow shear rate ranges (say, one decade), giving straight lines on linear or logarithmic plots. Other models such as Herschel-Bulkley (1926) and Casson (1959) for two or three decades and Sisko (1958) for four decades or more are better for wider shear rate ranges. For the widest shear rate range achievable in practice (that is, 10 -6 to 10 6 s -1 ), it is necessary to use a more complex model such as the Cross (1965) model or the Carreau (1968) model.
Of the numerous rheological models available in the literature, the most commonly-used for engineering applications are described here.

Newtonian model
This is the simplest of all flow curve models and is given by with η N being the Newtonian viscosity.

Power law model
This model originally proposed by de Waele (1923) and Ostwald (1925) is described by the two-parameter equation: ( where K is the consistency coefficient in units of Pas n and n is the power law exponent. This equation can be used to describe Newtonian behaviour when n = 1, shear-thinning behaviour when n < 1 or dilatant behaviour when n > 1. On a log-log plot, the model is a straight line with a slope of n. Values of n typically ranges from 0.2 to about 1.4. The further the value of n is from unity, the more non-Newtonian is the fluid.

Bingham plastic model
This two parameter model (Bingham, 1922) is described by (4) where τ yΒ is the Bingham yield stress and η Β is the Bingham plastic viscosity.

Herschel-Bulkley model
This model also known as the generalised Bingham plastic model, is a three parameter yield/power law model (Herschel-Bulkley, 1926), given by (5) where τ yHB is the Herschel-Bulkley yield stress. This equation describes viscoplastic behaviour when n < 1. Because power law (τ yHB =0; shear-thinning when n < 1 or dilatant when n > 1), Newtonian (τ yHB = 0 and n = 1) and Bingham plastic behaviour (n= 1) can be regarded as special cases, the model represents the flow behaviour of a wide range of fluids without being too difficult to handle mathematically.

Casson model
A popular alternative to the Herschel-Bulkley model is the theoretical two-parameter model of Casson (1959). This is given by (6) where τ yC is the Casson yield stress and τ C is the Casson viscosity.

Viscosity and flow curve measurement
Commercial viscometers and rheometers employ a wide range of geometries for viscosity and flow curve measurement. These can be grouped into two main types: rotational viscometers and tube viscometers.

Rotational viscometers
Rotational viscometers, which rely on rotational motion to achieve simple shear flow, can be operated either in the controlled rate or controlled stress mode. In controlled-rate instruments, there are two methods of applying the rotation and measuring the resultant torque. The first method is to rotate one member and measure the torque exerted on the other member by the test sample, whilst the second method involves the rotation of one member and measuring the resultant torque on the same member, The rotating member is either at constant speed which can be sequentially stepped or with a steadily-changing speed ramp. The resultant torque is measured by a torsion spring. In controlled-stress instruments, either a constant torque (which can be sequentially changed) or a torque ramp is applied to the member, and the resultant speed is measured. The more common geometries used in rotational viscometry are shown in Figure 6.

Tube viscometers
Tube viscometers are generally once-through batch devices consisting of either a horizontal or vertical length of precision-bored, straight tube through which the test fluid is passed at varying rates from a reservoir. The diameters of the tube can typically range from 1 to 5 mm. Essentially, there are two types of tube viscometer, the controlled flow rate and the controlled pressure, as shown in Figure 7.
In the controlled flow rate tube viscometer, a piston forces the fluid through a horizontal or vertical tube at a constant flow rate and the resultant pressure drop is measured. In the controlled pressure viscometer, compressed air (or nitrogen) is applied to drive the fluid through a horizontal or vertical tube and the resultant volumetric flow rate is measured.

Practical considerations
Flow curve measurements can be made using all of these geometries but there are several drawbacks for each geometry which need to be considered for each fluid under test and intechopen.com each specific application. Guidance on the use of these geometries for obtaining relevant viscometric data required for a particular application is given by Alderman andHeywood (2004a, 2004b). The simplest geometry (a) consists of a bob (inner cylinder) located in a cup (outer cylinder) with the test sample contained in the narrow annular gap between the bob and the cup. Other variations of the coaxial cylinder viscometer that are commonly used include (b) a bob rotating in a large container that approximates to the "infinite sea" situation (that is, the container to bob radius ratio is at least 10 and (c) the Moore-Davis double cylinder viscometer used for low viscosity fluids.

Rotating disc
This consists of a disc rotating in a large container (normally a 600 ml plastic beaker with an internal diameter of 97mm) of test material. The torque exerted by the test fluid on the rotating disc is measured as a function of rotational speed. Usually, the sample under test is contained between the exterior angle of a cone and a flat plate, (a). The axis of the cone is set normal to the plate with the cone apex touching the plate. Cone angles typically range from 0.25 to 1 o . However, for solid/liquid suspensions and emulsions, a cone with a truncated apex (b) is often used to minimise problems due to particle jamming.

Cone-and-plate
In controlled-rate instruments, either the cone or the plate is rotated at a fixed speed and the resultant torque via the cone is measured. In controlled-stress instruments, a fixed torque or a torque ramp is applied to the cone and the resultant speed is measured.

Parallel plate
The sample under test is held in the gap between two identical circular flat plates. The gap between the two plates can be varied, typically up to 5 mm for plates of about 25 mm radius.
In controlled-rate instruments, either the top or bottom plate is rotated at a fixed speed and the resultant torque is measured.
In controlled-stress instruments, a constant torque or a torque ramp is applied to the top plate and the resultant speed is measured.

Fig. 7. Types of tube viscometer
For the correct end-use of viscometric data in any processing application, the flow curve must be measured under appropriate conditions of temperature and shear. Hence, the shear rate (or shear stress) to be covered by the viscometer should be matched to that applicable to the particular flow process. With the range of viscometers commercially available, it is technically possible to measure viscosity over 12 orders of magnitude of shear rate from 10 -6 to 10 6 s -1 . However, most processing applications require viscosity data over no more than two or three orders of magnitude of shear rate. Shear rates typical of some flow processes can be found in Alderman and Heywood (2004a). There are quick and simple methods available for defining the shear rate range for any processing application, Alderman and Heywood (2004a). If a method for shear rate estimation in some processing applications is not available, a useful approach is to define the flow region of interest, determine differences in fluid velocity at two points across the region (this will often be the surface of some moving element such as a pump impeller or mixer agitator) and divide this velocity difference by the distance of separation between the two points. These typical shear rates can be used as a basis for choosing a viscometer.
As the equations for calculating the shear stress, shear rate and viscosity assume the flow in the viscometer is laminar, a check must be made to ensure the validity of the viscometric data. Details on how this is done are outlined in Alderman and Heywood (2004b).
www.intechopen.com Figure 8 summarises the key steps to ensuring the measurements of the flow curve are both accurate and relevant (Alderman and Heywood, 2004a;2004b).The need for accuracy may sometimes introduce additional laboratory experiments, particularly if end effect and/or wall slip errors are incurred. However, ensuring that only the viscometric data that are relevant to the application of interest are measured will minimise the overall effort. Having selected the most appropriate viscometer for the sample under test and the application of interest, the next step is to generate the flow curve.  (Alderman and Heywood, 2004b) www.intechopen.com

Rotational viscometers
With controlled rate rotational instruments, the flow curve measurement is usually done by carrying out at least three cycles of a fixed time where the speed is varied, either by a sequence of step speed changes or a steadily-changing speed ramp, between minimum and maximum values whilst measuring the resultant torque. With controlled stress rotational instruments, the torque is varied in a similar manner whilst the resultant speed is measured. The time over which the speed or torque is varied is left to the operator to decide. However, a good starting point would be the time that gives 30 s per step.
Repeated shear cycles on the test sample will enable one to determine whether the sample exhibits time-dependent flow behaviour such as thixotropy. If the up and down curves for the first and successive cycles coincide, the sample is undergoing steady-state shear. However, if hysteresis loops between the up and down curves are observed for each successive cycle, the sample is exhibiting time-dependent flow behaviour. In such cases, it is advisable to repeat the experiment with the speed (or torque) held constant until the torque (or speed) attains a steady value before changing the speed (or torque) to the next value. This will yield an equilibrium flow curve in which the up and down curves coincide.
The average of the torque versus speed data at which the up and down curves coincide is the first step of the calculation procedure for obtaining the corrected flow curve. Care must be taken to ensure that the flow curve is not affected by the four error sources: secondary/turbulent flow, viscous heating, end effect and wall slip. Further details of this procedure can be found in Alderman and Heywood (2004b) for the appropriate viscometric geometry used.

Tube viscometers
With controlled-rate tube viscometers (Type A in Figure 7), the flow curve is obtained by carrying out pressure drop measurements at different constant flow rates. For each measurement, a new sample is often required due to the small capacity of the sample reservoir. With controlled-stress tube viscometers (Type B in Figure 7), the flow curve is generated by measuring the flow rate as a function of pressure drop on the sample. The pressure drop versus flow rate data is the first step of the calculation procedure for obtaining the corrected flow curve. Care must be taken to ensure that the flow curve is not affected by the four error sources: secondary/turbulent flow, viscous heating, end effect and wall slip. Further details of this procedure can be found in Alderman and Heywood (2004b).

Viscosity and flow curve interpretation for engineering design
Having completed the calculation procedure for the corrected flow curve (and hence the corrected viscosity curve), the data may be amenable to a single curve fit (Brown & Heywood, 1991). Sometimes, because of considerable scatter in the data, it may be more appropriate to construct at least two curves: a mean curve obtained from regression analysis using all the data and an upper bound curve obtained from regression analysis using (τ, ) data selected from the curve that was initially drawn by eye. The upper bound curve would normally represent the worst case for many engineering applications and would lead to a conservative design. Further factors can cause difficulties in attempting to draw a single flow curve through the data. These factors include the use of two or more different γ  www.intechopen.com viscometric geometries which may give differing degrees of phase separation during shear, sample variability taken from the same batch, and uncorrected errors associated with the use of any viscometric geometry.
For engineering design, the choice of flow curve model is limited by the design method to be used. For example, if the flow curve is to be used for pipeline design, the choice can be made from a number of models including Newtonian, power law, Bingham plastic, Cassonand Herschel Bulkley models. However if the flow curve is to be used for agitation in stirred tanks, the choice is either the Newtonian or the power law model. It is often not immediately obvious from the data which of the flow models should be selected. A decision will need to be made on which model to use. The following approach is suggested: i. Plot all the (τ, ) data on linear axes and separately, on double logarithmic axes. This is to assess the suitability of the Newtonian, Bingham plastic and power law models. ii. If there is considerable scatter in the data, decide by eye or from the correlation coefficient obtained by linear regression analysis whether a straight line through the linear or the log-log plot gives the better representation. Similarly decide for the upper bound curve. If one of these alternatives is acceptable, the use of the Herschel-Bulkley model is probably not warranted. iii. If neither of these alternatives appears satisfactory because there is significant curvature of the data on both linear and log-log plots, the following can arise: a. If there is data curvature on the log-log plot with the slope of the curve increasing with shear rate axis and if the linear plot does not produce a straight line then the Herschel-Bulkley model should adequately describe the data. b. If there is data curvature on the log-log plot and the slope of the curve decreasing with shear rate axis, then the use of the Herschel-Bulkley model is inappropriate as this implies a negative yield stress parameter. However, a curve fit is possible and would result in a negative value for the yield stress parameter. Force-fit either a Bingham plastic or power law model to the data.
Estimates need to be made for the parameters defined in the flow models. As the Herschel-Bulkley model can be reduced to the Newtonian, power law and Bingham plastic models, a least squares regression analysis can first be performed on the (τ, ) data to obtain τ yHB , K and n. It may then be possible to simplify the model by setting the τ yHB to zero if the estimate is close to zero and/or setting n to 1 if the estimate is close to unity.
Two methods are commonly used when carrying a regression analysis on the (τ, ) data (Heywood & Cheng, 1984): 1. a non-linear least squares regression on unweighted data, 2. a non-linear least squares regression on weighted data.
In Method 1 it is assumed that the error e lies in τ: whereas in Method 2, the error is assumed to lie in ln (τ -τ yHB ): Standard non-linear regression software packages can be used in either case. Alternatively, non-linear regression can be performed using Microsoft Excel via the 'Solver' tool (Roberts et al., 2001). Both methods will provide sets of τ yHB , K and n estimates which give viscometric data predictions to ± 2% of the original data within the original shear rate range. Outside this shear rate range, agreement can be poor.
Examples of the extrapolated flow curves for the two sets of parameters obtained in the two regression methods are shown in Figure 9. Provided the relevant shear rate/shear stress window for the application is covered by the viscometer, either of the two regression methods can be used. However, the method that gives the most uniform data spread should be used. Fig. 9. Example of extrapolation of predicted (τ, ) data outside the experimental range for an 8.0% digested sewage sludge (Heywood and Cheng, 1984)

Pipe flow fundamentals
The basic relationships for design in laminar, transitional and turbulent pipe flow are obtained by integration of the constitutive rheological relationship, over the cross-sectional area of the pipe.

Laminar flow
Equations for the design of laminar pipe flow can be derived by integrating Eqn. (5) over the circular pipe geometry (Govier & Aziz, 1972). Because of the yield stress, a central solid plug is formed where the point shear stress is less than the yield stress, as shown in Figure 10. The radius of the plug is determined from the linear shear stress distribution and the yield stress as (9) In the annular region where the point shear stress exceeds the yield stress, the point velocity u is obtained by integration of the constitutive rheological relationship, and applying the noslip assumption at the pipe wall The plug velocity u plug is obtained at the point where the applied shear stress is equal to the yield stress as These two velocity distributions are integrated over the pipe cross-sectional area appropriate to each, in order to yield the total volumetric flow rate Q as This relationship can be used for laminar pipe flow design.

Transitional flow
The approach used here for the prediction of the transition from laminar to turbulent flow is the modified Reynolds number Re 3 (Slatter, 1995;Slatter, 1999). This approach predicts a www.intechopen.com laminar to turbulent transition in the Reynolds number region of 2100. This approach was specifically developed to place emphasis on the viscoplastic nature of the material (Slatter, 1995). Using the fundamental definition that Re ∝ inertial/viscous forces, the final expression is (13) As shown in Figure 10, in the presence of a yield stress the central core of the fluid moves as a solid plug which fundamentally affects the stability of flow (Slatter, 1995(Slatter, , 1999. The unsheared plug is treated as a solid body in the centre of the pipe. The flow that the plug represents must be subtracted as it is no longer being treated as part of the fluid flow. The corrected mean velocity in the annulus V ann is then obtained as follows:- and Q plug = u plug A plug The sheared diameter, D shear , is taken as the characteristic dimension because this represents the zone in which shearing of the fluid actually takes place, and it is defined as and D plug = 2r plug (17) These relationships can be used for pipe flow design to determine the transition from laminar to turbulent flow.

Turbulent flow
The approach used here is the particle roughness turbulence approach (Slatter, 1996(Slatter, , 1999b(Slatter, , 2011. The point of departure of this approach is the classical logarithmic velocity distribution * =A ln +B The value of A is taken as the inverse of the von Karman universal constant, A=1/χ = 2.5. B is the classical roughness function (Schlichting, 1960). Integrating and rearranging we get the mean velocity V as * = ln + B − .7 For smooth wall turbulent flow, this reduces to For fully developed rough wall turbulent flow, this reduces to * = . ln + .7 which will yield a constant value for the Fanning friction factor, f √ = . 7 log .

Minor losses in pipe systems
Head losses, in addition to those due to straight pipe friction, are always incurred at pipe bends, junctions, contractions, expansions and valves. These additional losses are due to eddy formation generated in the fluid at the fitting. In the case of long pipelines of several kilometres, these local losses may be negligible, but for short pipelines they may be greater than the straight pipe frictional losses (Chadwick & Morfett, 1993). A general theoretical treatment for local head losses is not available, but it is usual to assume rough turbulence (where the friction factor is independent of the Reynolds number) since it leads to a simple equation (Chadwick & Morfett, 1993). The prediction of losses in pipe fittings is either based on (King, 2002): a. The fitting will contribute to the energy dissipation an amount equivalent to an additional length of pipe that is calculated as a multiple of the pipe diameter. b. The kinetic energy is dissipated as the fluid flows through the fitting and the loss is calculated in terms of the number of velocity heads that are lost.
Here, the losses in pipe fittings will be expressed as the number of velocity heads lost.
On dimensional grounds, the head loss in a fitting will depend upon the fluid velocity, fluid properties and the geometry of the fitting as follows (Edwards et al., 1985): This can be expressed as a function of the velocity energy head given by (24) where h fitt is the local head loss and k fitt is the fitting loss coefficient.
Since the pressure drop across the fitting, Δp fitt is given by Δp fitt =ρgh fitt , Eqn. (24) can be rewritten as The loss coefficient, k fitt , is the non-dimensionalised difference in overall pressure between the ends of two long straight pipes when there is no fitting and when the real fitting is  (Miller, 1978). The flow lengths over which pressure losses occur start from a few diameters upstream to several pipe diameters downstream of the actual length of the fitting. This is known as the region of influence or interference (see Figure 11). The pressure loss across the fitting Δp fitt should be measured across this region. It can be the measured static pressure drop (Δp s ) or the total pressure that is Δp tot = Δp s +½ρV 2 . It is therefore important to state whether k fitt is based on the static or total pressure (Miller, 1978). If there is a change in pipe diameter, the convention is to use the higher mean flow velocity (V) of either the upstream or the downstream pipe. i. the actual length of the centreline of the entire system is taken; ii. the lengths of the individual pieces of pipe that are actually straight are summed up; iii. the distances between the intersections of the extended centrelines of the successive straight pipes are added.

Turbulent flow loss coefficient
For turbulent flow through fittings, with the exception of bends, the loss coefficient is independent of the Reynolds number because inertia forces dominate. Experimental work showed that this is true for Newtonian fluids (Miller, 1978, Crane, 1999 and non-Newtonian fluids (Edwards et al., 1985, Ma 1987, Turian et al., 1998.

Laminar flow loss coefficient
For laminar flow, the loss coefficient is inversely proportional to the Reynolds number (Hooper, 1981;Edwards et al., 1985;Ma, 1987;Pienaar, 1998) where K 1 is the laminar flow loss coefficient constant and is a characteristic of a specific fitting involving its dimensions.

Generalised correlation for loss coefficient
Hooper (1981) provided a generalised correlation for the determination of the loss coefficient spanning Reynolds number from laminar to turbulent flow (27) where K 1 and k turb are determined experimentally and D i is the pipe diameter in inches (King, 2002).

Determination of loss coefficients
The energy losses across a fluid-conveying conduit in a fluid are normally accounted for using the mechanical energy balance: where subscripts 1 and 2 refer to the upstream and downstream conditions respectively, V is the mean flow velocity, z is the elevation from the datum, is the kinetic energy correction factor and p is the static pressure and where there are N sources of energy loss (Edwards et al., 1985). Each term in the expression represents energy per unit weight of fluid, known as energy head or head loss, and is a statement of the law of conservation of energy as applied to fluid flow. The head loss is in units of metres. For the case of a head loss in a fitting, the energy equation may be rewritten as: where h 1 and h 2 refer to the friction head loss in the straight pipe upstream and downstream of the fitting and h fitt refers to the head loss in the fitting. The head loss, h, in the straight pipe can be calculated from In practice, it is difficult to measure the pressure drop across the fitting only (Ward Smith, 1976) to distinguish between the incompletely developed and the fully-developed flow region prior to and after the fitting. One method is to extrapolate the fully developed pressure gradient to the fitting centreline if the pressure is measured along the length of the two pipes. The second method is to measure the total pressure drop across the system by using two pressure taps only. The latter is experimentally a cheaper method, as only one pressure transducer is required. There are difficulties associated with both methods that could influence the results obtained to some extent. For the pressure gradient to be determined, it is required to select the points that are in the fully developed friction gradient region, but this is not always an easy task, as the distance of interference changes with Reynolds number (Pal & Hwang, 1999). For the total pressure drop method on the other hand, one needs to ensure that the additional losses are accounted for by ensuring that there is significant length of straight pipe. The difficulty is often encountered that to determine the loss in the fitting, two large numbers are being subtracted to obtain a very small number, resulting in significant errors or negative results (Sisavath, 2002). The following was found when data obtained for Newtonian lubrication oil flowing through sudden contractions was analysed using the pressure grade line approach as well as the total pressure drop approach to obtain values of kturb and K 1con and compared to results found in literature as shown in Figure 12.  (Pienaar & Slatter, 2004) Between Re = 10 -10 000, the fact that all the data points are used for extrapolation or that some data are selected gives little difference in the loss coefficient calculated. It is clear that the range of data selected for analysis of results has a significant effect at Reynolds numbers less than 10 and greater than 10 000. Good agreement was found between this work using all data and that of Jadallah (1980) and the prediction of ESDU (1989).

Loss coefficient, k con
This work all data This work selected data McNeil andMorris (1995) ESDU (1989) Jadallah 1980Holland (1973) Hwang 1999Pienaar 1998_glycerol Pienaar 1998_water Ma 1987Ma (1987 calc This work_glycerol f = exp This work glycerol f = 16/Re • At Re > 10 000, the results were significantly lower than predicted results and it was obvious that in this range the data closed to the contraction plane had to be excluded from the pressure gradient analysis. • At Re < 10, good agreement was found between this work using selected data and the prediction of McNeil and Morris (1995).

•
Discrepancies between K 1 values are often due to the fact that results are not obtained at Reynolds numbers less than 10 in the purely viscous range before onset of turbulence.
It is evident that it is not only the physical experimental measurements that contribute to the large scatter of results, but also the analytical approach applied to the experimental results to obtain loss coefficients. It is important to analyse experimental results in various ways by assessing experimental data very carefully.

Dynamic similarity
Complete analytical solutions for engineering problems involving the flow of real fluids is seldom attainable and experiments on models of different physical size are often a necessary part of the design process. This investigation is typical of this approach. In order to correctly interpret the qualitative and quantitative data obtained from such experiments, it is necessary to understand the relationship between models of different size. The concepts of dimensional analysis and physical, geometric, kinematic and dynamic similarity are introduced as they apply to the specific problem of modelling and data interpretation of fittings of different size. Dimensional analysis enables the magnitudes of individual quantities relevant to a physical problem to be assembled into dimensionless groups, often referred to by name. The dimensionless group of specific interest here is the Reynolds number. These groups assist in the interpretation of model studies by ensuring that the conditions under which tests and observations take place at one size fitting are the same as those at the other size fitting.
Physical similarity, like dimensional analysis, helps to ensure that the conditions under which tests and observations take place at one scale are the same as those on another scale. The models at different scale are said to be physically similar in respect of specified physical quantities (eg, velocity), when the ratio of corresponding magnitudes of these quantities between the two systems are everywhere the same. For any comparison between models, the sets of conditions associated with each must be physically similar.
Geometric similarity is similarity of shape. The requirement is that any ratio of length in one model to the corresponding length in another model is everywhere the same. This ratio is referred to as the scale factor. Geometric similarity is the first requirement of physical similarity. Kinematic similarity is similarity of motion and requires similarity of both length and time interval.
Dynamic similarity is similarity of forces. Since there may be several kinds of forces acting on a fluid particle, it is usually impossible to satisfy dynamic similarity for all of them simultaneously. The justification for comparing observations from one model flow system to another is that the fluid behaviour in both systems is similar thus implying kinematic similarity. Geometric similarity alone does not imply dynamic similarity. The requirement for kinematic similarity is to have both geometric and dynamic similarity. This produces geometric similarity of flow patterns and it is this which is of prime importance in this study.

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It was shown by Edwards et al. (1985) and Fester and Slatter (2009) that if geometric similarity is maintained, the Reynolds number can be used to establish dynamic similarity in globe and gate valves. Although Edwards et al. (1985) initially indicated that for turbulent flow in globe valves, the turbulent loss coefficient is geometry dependent, this was later found to be due to the globe valves not being geometrically similar. Fester and Slatter (2009) found that the turbulent loss coefficient is in fact independent of the valve size for the geometrically similar globe valves tested. This was found to be in agreement with Edwards et al.'s (1985) findings for gate valves. It will only depend on the valve opening for carefully machined valves. However, results for diaphragm valves agreed with Edwards et al.'s statement that for significant changes in cross-sectional areas, the result becomes size dependent. The viscous force dominated range at Re < 10 is not affected by valve size. In fact, the K 1 value obtained for non-Newtonian fluids is in excellent agreement with that provided by Hooper (1981). As the Reynolds number increases beyond the critical Reynolds number as inertia forces begin to dominate, the loss coefficient is sensitive to geometry and valve size. Results for valves from different manufacturers are remarkably different in this region and this is directly related to the flow path. Furthermore, the valves were rubberlined, and this influenced the actual final diameter of the valve. Miller (1978) presented general idealized curves of the laminar to turbulent transition region as shown in Figure 13. The transition is not always sharp since it can be the entire region from where the loss coefficient starts to deviate from the trend of being inversely proportional to the Reynolds number to where it becomes independent of the Reynolds number. Sometimes a minimum is reached before the loss coefficient increases and then takes a constant value in turbulent flows. The transition region starts where the flow upstream of the fitting is laminar, but downstream turbulence is introduced (Jameson & Villemonte, 1971). Since the Stokes flow region is between 1 < Re < 10, it is important that tests must be conducted in this range to ascertain the onset of transitional flow. Various Fig. 13. Trends in loss coefficients in the laminar to turbulent transition region (Miller, 1978) www.intechopen.com conflicting transition Reynolds numbers that have been reported is mainly due to the different definitions used for onset of transitional flow. It has been defined as either the intersection between the straight lines for laminar flow and turbulent flow (Ma, 1987) or as the point where the data starts to deviate from the laminar flow line (Fester & Slatter, 2009). The latter always occur at Reynolds numbers lower than that predicted by the intersection method. For short orifices, the ratio of the Reynolds number from the onset of deviation from laminar flow to that of fully developed turbulent flow was found to be ≈0.008 for ratios ranging from 0.2 to 0.7. For sudden contractions and long orifices, this minimum value is not as pronounced as for short orifices as shown in Figures 14 to 16.

Loss coefficient data
Loss coefficients for both laminar and turbulent flow of Newtonian and non-Newtonian fluids through various types of fittings are provided for engineering design calculations.

Contractions and expansions
For sudden contractions and expansions where is the ratio of the downstream (d d ) to upstream (d u ) pipe diameter, K 1 values for non-Newtonian fluids were found to be similar to those found for Newtonian fluids. Here, the non-Newtonian behaviour of the fluid was accounted for by using the appropriate Reynolds number for the fluid. This was the conclusion drawn by Edwards et al.(1985), Ma (1987), Pienaar (1998), Pal and Hwang (1999) and Fester et al. (2008). Mika (2011) was, however, unable to establish dynamic similarity for laminar flow of ice slurries of various concentrations in sudden contractions. Tables 1 and 2 show a comparison of the results obtained by various workers for sudden contractions and expansions respectively. In general, there was much better agreement between values in turbulent flow than those in laminar flow for which no quantitative agreement was found amongst the work of different researchers or correlations to predict losses through sharpedged contractions and expansions.

Valves
Valves form an integral part of slurry pipework systems. These fall within the category of fittings that influence slurry flow in pipework by opening, closing, diverting, mixing or partially obstructing the flow passage (Whitehouse, 1993). An isolation (on-off) valve is a valve designed for use in either the fully open or closed position. Those suitable for slurry service include knife gate, parallel gate, diaphragm, pinch, plug, ball, butterfly and rotating disc (Alderman & Heywood, 1996).
A regulating (throttling) valve is a valve designed for use in all positions between fully open and fully closed. Those suitable for slurry service include globe, diaphragm, pinch and segmented ball (Alderman & Heywood, 1996). Work on frictional pressure losses arising from flow through valves has been carried out with Newtonian slurries using both isolation and regulating valves. Initially, work with non-settling, non-Newtonian slurries appears to have been restricted to gate and globe valves (Edwards et al., 1985;Turian et al., 1998;Pal & Hwang, 1999). This work was extended to diaphragm valves by Mbiya et al. (2009) and Kabwe et al. (2010).

Globe and gate valves
A summary of loss coefficient data for gate and globe valves are provided in Table 3 ranging from 12.5 mm to 40 mm. Figure 17 shows experimental data obtained for 12.5, 25, 40 and 65 mm globe valves from two different manufacturers. A range of Newtonian and non-Newtonian fluids were tested for Reynolds numbers covering from 0.05 to 1000000 obtained from three different test rigs (Fester & Slatter, 2009). In laminar flow, good agreement was obtained with those obtained by Hooper for a standard globe valve, but in turbulent flow, the predicted values for a 1 inch globe valve approximated those for the 65 mm valve. The laminar to turbulent transition for the 12.5, 25 and 40 mm valves was found to be smooth whereas a minimum was obtained for the 65 mm valve.  (Fester & Slatter, 2009)

Diaphragm valves
Extensive experimental work was conducted on straight-through diaphragm valves from two manufacturers (Kabwe et al., 2010). The nominal valve sizes were 40, 50, 65, 80 and 100 mm. The fluids tested were water, carboxymethyl cellulose solutions and kaolin suspensions. It was observed that at Reynolds numbers below 10, the loss coefficient is independent of the valve opening for valves from both manufacturers. This purely laminar flow regime can be modelled using a K 1valve constant of 1000. This is in agreement with the value provided by Hooper (1981). However, in the fully turbulent regime this loss coefficient for the Natco valves were found to be higher than those for Saunders valves. This may be attributed to the more tortuous flow path of the Natco valve compared with that of where λ Ω is k valve at 100% opening and θ is the valve opening. The results obtained for the valves are given in Table 4.

Orifices
Loss coefficient data for orifices are provided in Table 5 and 6. Johansen, in 1930, initiated the first detailed studies on sharp edged concentric orifices but as recent as 2007 the lack of pressure loss coefficient data for ratios of 0.2, 0.3, 0.4, 0.5 and 0.7 in literature for long and short orifices was identified (ESDU, 2007). Little work has been done with non-Newtonian fluid despite their importance in the field of polymer processing, flow of petroleum products, biomedical engineering, biochemical engineering, food processing, and mineral processing plants. In such applications, the flow remains laminar even at large flow rates (Bohra , 2004). Correlations found in literature are generally for turbulent flow regimes for which the loss coefficients mainly depends on the geometry of the orifice, practically independent of Reynolds number.

Long orifices
In 1972, Lakshmana Rao and his co-workers obtained the pressure loss coefficients for laminar flow in five sharp square-edged long orifices with a constant ratio of 0.2 whilst varying the thickness to diameter ratios from 0.48 to 10.11. Pressure loss coefficients in turbulent flow for orifices with ratios of 0.36, 0.4, 0.5 and 0.7 with aspect ratios of 4, 4, 5 and 5 respectively are compared with those of Ward-Smith (1971) and Idel'chik et al. (1994). Excellent agreement was found between experimental work and turbulent flow correlations from Ward-Smith (1971) and Idel'chik et al. (1994), with maximum difference between the experimental data models were 9.87% and 12% respectively for the maximum beta ratio of 0.7. This difference was well within the experimental error obtained. The comparison shows clearly that correlations published by Ward-Smith (1971) and Idel'chik et al. (1994) can be used from Re > 1000, although the two models has an applicability range from Re > 10 4 .
Due to lack of correlations to predict losses through long orifices for laminar flow, the correlations of Hasegawa et al. (1997) and Bohra (2004), although limited to = 0.1 and 0.137 respectively, were evaluated at the higher orifice diameter ratios tested in this work. Hasegawa et al. (1997) shows good agreement in laminar flow for the Stokes flow region, Re < 10, for = 0.5. It is however unable to predict turbulent flow. Bohra (2004) approximates turbulent flow data very well for all cases except for = 0.7. However, there is good agreement in laminar flow for = 0.7. The comparison ( Figure 18) also revealed that an improvement of the models is required to predict losses accurately over a wide range of laminar and turbulent flow. No suitable correlation was found in the literature to predict pressure losses through long square edged orifices from laminar to turbulent flow regimes. The loss coefficient data obtained from the experimental work  is given in Table 5.

Short orifices
Although the work on orifices started in the early 1900s, it was mainly for determining discharge coefficients (Johansen, 1930;Medaugh & Johnson, 1940). A comparison of pressure loss characteristics of different geometries orifices and nozzles was done by Alvi et al. (1978). They found that the flow characteristics of orifices can be divided into three regimes: fully laminar region, re-laminarising region and turbulent region. Lakshmana Rao et al. (1977) investigated the critical Reynolds number for orifice and nozzle flows and found that the critical Reynolds number approached a constant value for low value of orifice or nozzle diameter to pipe diameter ratio. The work done on flow through orifice plate was carried out using Newtonian fluids. In such applications, the flow remains laminar even at large flow rates (Bohra et al., 2004). Edwards et al. (1985) used aqueous solutions of carboxymethylcellulose and suspensions of china clay in water through orifice plates.
www.intechopen.com Significant differences were found between the experimental results of Edwards et al. (1985) and those of Ntamba (2011) as shown in Table 6. However, the results of Ntamba (2011) shown in Figure 19 are in good agreement with that of Lakshmana Rao et al. (1977).  In order to illustrate the effect that the fittings loss has in laminar viscoplastic flow, a simple system consisting of 10 m of straight 50 mm ID pipe and 5 fittings -the loss coefficient of the above diaphragm valve in laminar flow is k v =946/Re 3 , and for turbulent flow is constant at k v = 2.5 -is set and analysed. The fluid used for the analysis is a viscoplastic paste (τ y = 100 Pa, K = 1 Pa.s, relative density = 1.5 and n = 1). These values were chosen so as to present a relatively simple viscoplastic rheology which would yield laminar flow in a 50 mm pipe at 3 m/s.
The objective of the analysis was to obtain the head loss as a function of volumetric flow rate. The operating flow rate considered was 0.006 m 3 /s which corresponded to an operating average velocity V of 3 m/s in a 50 mm pipe.
Since the value of the laminar flow loss coefficient is not available to most designers, the effect of incorrectly using the turbulent -constant -value for design in laminar flow will be highlighted. Figure 20 shows the fittings head losses in both laminar and turbulent flow. Figure 20 shows three principal differences between the fittings head losses in both laminar and turbulent flow: 1. The significant contribution of the laminar fittings loss to the start-up static head (at Q = 0). 2. The very different shape presentation between laminar and turbulent flow. 3. The significant difference in magnitude which arises. www.intechopen.com

Start-up static head
In this example, the contribution of the laminar fittings loss to the start-up static head (at Q = 0) is 6.5 m, as can be seen on the ordinate of Figure 20. Inspection of Eqn. (24) might lead one to expect a zero result at Q=0, as portrayed by the turbulent locus in Figure 20.
However, it must be understood that this non-zero value is a result of the combined effects of Eqns. (13), (24) and (26), and is a sign of the presence of a yield stress.

Shape presentation
The inherently parabolic shape of the turbulent locus in Figure 20 is a direct consequence of the quadratic form of Eqn. (24) and the constant value of the loss coefficient k fitt =2.5 for this valve type in turbulent flow.
Equally, the inherently viscoplastic shape of the laminar locus in Figure 20 is a direct consequence of the hyperbolic form of Eqn. (26) combined with the emphasis of the role of the yield stress in Eqn. (13). Figure 20 shows that there are considerable differences in the magnitude of the fittings Head losses in laminar and turbulent flow. A direct comparison of these magnitudes is shown in Figure 21.

Pump power predictions
Whilst these three issues discussed above are of importance, the issue of primary practical interest for paste piping design is the error in power prediction, if the design is performed using the incorrect approach . This situation is summarised for this example in Figure 22. Figure 22 shows the pump operating points for both the laminar and turbulent design cases, as derived from the practical design case example presented above.
The three possible operating points arising from the different design approaches to the practical design case example are highlighted in Figure 22 and designated Point A, Point B and Point C. The coordinates and resulting motor power (brake power) requirements from each of these three operating points are presented in Table 7.   Table 7. Pump operating point coordinates from Figure 22 If the design is performed using the incorrect approach , the (incorrect) design operating point will be at Point A for this example, as shown in Figure 22 and Table 7.
If the designer follows the information published by the authors (Fester et al., 2007;Mbiya et al., 2009) , the (correct) design operating point will be at Point C for this example, as shown in Figure 21 and Table 7.
This presents at least two problems of profound practical importance:-1. If the designer follows the first (incorrect) approach, then Figure 22 shows that -in reality -the pump will actually operate, not at Point A, but at Point B, at pump speed N 1 . Point B presents an operating flow rate penalty exceeding 20% whilst being almost 10% under-powered, as shown in Table 7. 2. In order to achieve the desired flow rate, the operating point will need to move from Point B to Point C, i.e. the speed would need to be significantly increased to pump speed N 2 . This increase in pump speed will require 50% more brake power, as shown in Figure 22 and Table 7.
The important practical message here is that the pump motor would be 50% undersized.
All of these issues are further exacerbated by the shallow intersection angle which the laminar flow system curve presents to a centrifugal pump curve, as shown Figure 22. This will result in unstable operation, as small changes in head (i.e. small changes in material consistency) will result in significant changes in operating point flow rate.
As indicated by other laminar flow investigations over several decades (e.g. Edwards et al., 1985;Fester et al., 2008), all fittings show the k v values in laminar flow to be significantly greater than those in turbulent flow.

Conclusions
Minor losses are important in the efficient design of pipelines in laminar flow. The loss coefficients in laminar flow for all fittings are orders of magnitude greater than those in turbulent flow. The purely viscous driven flow regime is primarily at Re < 10, after which turbulence will be introduced that is dependent on the geometry of the fitting. For some fittings, the minima value obtained are very pronounced, especially for larger orifice ratios and larger valve sizes. For sudden contractions and long orifices, a smoother transition is observed. Dynamic similarity can be established for geometrically similar fittings using a Reynolds number that can accommodate the rheological parameters of the fluid. Pump sizing estimates for shorter lengths of pipelines with a number of fittings operating in laminar flow can be underpredicted by up to 50%. The significance of the impact on pump sizing would therefore appear to be representative of other fittings in general, but focussed research to substantiate this is required.